Mathematical Sciences: Presidential Young Investigator Award

数学研究的主要焦点是 总统青年调查员将在偏微分 由几何问题产生的方程。 了方程 典型地为椭圆和抛物线型的非线性。 有一类问题涉及所谓的曲线问题 缩短。 问题是如何描述 一系列平面曲线的行为，这些曲线的演变使得它们的 法向速度与曲率一致。 更复杂 当法向速度一般取决于 在曲率和时间上的方式;后一种情况发生在 二维相变模型。 曲线缩短问题的解决方案已被证明， 在某些情况下会产生奇点（爆破）。 的 （适当地重新缩放）奇异部分的渐近形状确实 不依赖于原始数据。 数值模拟表明 曲率的实际放大率是 时间的重对数 将在确定 如果这一观察是正确的， 负责估算中的对数项。 工作也将在确定周期轨道与 余切丛上Hamilton系统的规定能量 光滑致密的表面。 假设哈密顿量是凸的 甚至在每根纤维上。 这项工作的一个特别目标是 建立能量面上周期轨道的存在性 其在流形上的投影与 给定周期轨道的规定次数。 这样的结果 可以被看作是庞加莱-伯克霍夫模型的全球模拟 定理 相关工作将研究 二维哈密顿系统的解。 估计 连接数可以被看作类似于测量 纯量抛物型偏微分方程根的个数 方程

The primary focus of the mathematical research of this Presidential Young Investigator will be on partial differential equations arising from geometrical problems. The equations are typically nonlinear of elliptic and parabolic type. One class of problems involves what is known as curve shortening. The question is one of describing the long time behavior of a family of plane curves which evolve so that their normal velocity coincides with their curvature. More complicated situations arise when the normal velocity depends in a general way on the curvature and time; the latter case occurring in models of phase transitions in two dimensions. The solution to a curve shortening problem has been shown to develop singularities (blow-up) under certain circumstances. The asymptotic shape of the (suitably rescaled) singular part does not depend on the initial data. Numerical simulations suggest that the actual blow-up rate of the curvature is a function of the iterated logarithm of time. Work will be done in determining if this observation is true and in seeking the mechanism responsible for the log-term in the estimate. Work will also be done in determining periodic orbits with prescribed energy of Hamiltonian systems on cotangent bundles of smooth compact surfaces. The Hamiltonian is assumed to be convex and even on each fibre. A particular goal of the work is to establish the existence of periodic orbits on energy surfaces whose projections on the manifold intersect the projection of a given periodic orbit a prescribed number of times. Such a result may be regarded as a global analog of the Poincare-Birkhoff theorem. Related work will study the linking of parts of solutions of Hamiltonian systems in two dimensions. Estimating linking numbers can be viewed as analogous to measuring the number of roots of a scalar parabolic partial differential equation.